Some two-dimensional extensions of Bougerol’s identity in law for the exponential functional of linear Brownian motion

The main result is a two-dimensional identity in law. Let (B t ,L t ) and (β t ,λ t ) be two independent pairs of a linear Brownian motion with its local time at 0. Let A t =∫ 0 t exp(2B s )ds. Then, for fixed t, the pair (sinh(B t ),sinh(L t )) has the same law as (β(A t ),exp(-B t )λ(A t )), and also as (exp(-B t )β(A t ),λ(A t )). This result is an extension of an identity in distribution due to Bougerol that concerned the first components of each pair. Some other related identities are also considered